Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis

We study the eigen-structure of the penalty matrix arising from cubic smoothing splines on equally spaced knots. Using purely matrix-theoretic arguments, we show that its positive eigenvalues are simple, that the associated eigenvectors alternate between even and odd, and that the eigenvector for the $k$th largest eigenvalue has exactly $k+1$ sign changes. The approach provides a direct and transparent alternative to existing variational proofs of the oscillation property. These results show that equally spaced knots support a spline basis with both a parity structure and an oscillation pattern.